Improved inequalities between Dirichlet and Neumann eigenvalues of the biharmonic operator

Lotoreichik, Vladimir

doi:10.1090/proc/16749

Improved inequalities between Dirichlet and Neumann eigenvalues of the biharmonic operator
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by Vladimir Lotoreichik;

Proc. Amer. Math. Soc. 152 (2024), 2571-2584

DOI: https://doi.org/10.1090/proc/16749

Published electronically: April 25, 2024

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Abstract:

We prove that the $(k+d)$-th Neumann eigenvalue of the biharmonic operator on a bounded connected $d$-dimensional $(d\ge 2)$ Lipschitz domain is not larger than its $k$-th Dirichlet eigenvalue for all $k\in \mathbb {N}$. For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the $(k+d+1)$-th Neumann eigenvalue of the biharmonic operator does not exceed its $k$-th Dirichlet eigenvalue for all $k\in \mathbb {N}$. In particular, in two dimensions, this special class consists of domains having an axis of symmetry.

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